This question is inspired in part by this answer of Bill Dubuque, in which he remarks that the fairly common belief that Kummer was motivated by FLT to develop his theory of cyclotomic number fields is essentially unfounded, and that Kummer was instead movitated by the desire to formulate and prove general higher reciprocity laws.

My own (not particular well-informed) undestanding is that the problem of higher reciprocity laws was indeed one of Kummer's substantial motivations; after all, this problem is a direct outgrowth of the work of Gauss, Eisenstein, and Jacobi (and others?) in number theory. However, Kummer did also work on FLT, so he must have regarded it to be of some importance (i.e. important enough to work on).

Is there a consensus view on the role of FLT as motivating factor in Kummer's work? Was his work on it an afterthought, something that he saw was possible using all the machinery he had developed to study higher reciprocity laws? Or did he place more importance on it than that?
(Am I right in also thinking that there were prizes attached to its solution which could also
have played a role in directing his attention to it? If so, did they actually play any such role?)

Presumably in due time Lemmermeyer will give a good answer to the question. In the meantime I suggest looking at section 2 of his paper "Jacobi and Kummer's ideal numbers". It is in Abh. Math. Semin. Univ. Hamburg 79 (2009), 165--187. If you have an institutional affiliation you may be able to view it as a .pdf file at springerlink.com/content/v82kw611k46t30ng.
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KConradAug 7 '10 at 3:05

1

Ah, whoops, I see Lemmermeyer's paper is already linked in the answer that Matt links to in his question!
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KConradAug 7 '10 at 3:07

3 Answers
3

When Kummer started working on research problems,he tried to solve what became known as "Kummer's problem", i.e., the determination cubic Gauss sums (its cube is easy to compute). Kummer asked Dirichlet to find out whether Jacobi or someone else had already been working on this, and to send him everything written by Jacobi on this subject.
Dirichlet organized lecture notes of Jacobi's lectures on number theory from 1836/37 (see also this question), where Jacobi had worked out the quadratic, cubic and
biquadratic reciprocity laws using what we now call Gauss and Jacobi
sums.

In his first article, Kummer tried to generalize a result due to Jacobi, who had proved (or rather claimed) that primes $5n+1$, $8n+1$, and $12n+1$ split into four primes in the field of 5th, 8th and 12th roots of unity. Kummer's proof of the fact that primes $\ell n+1$ split into $\ell-1$ primes in the field of $\ell$-th roots of unity (with $\ell$ an odd prime) was erroneous, and eventually led to his introduction of ideal numbers.

After Lame (1847) had given is "proof" of FLT in Paris, Liouville had
observed that there are gaps related to unique factorization; he then
asked his friend Dirichlet in a letter whether he knew that Lame's assumption was valid (Liouville only knew counterexamples for quadratic fields). A few weeks later, Kummer wrote Liouville a letter. In the weeks between, Kummer had looked at FLT and found a proof
based on several assumptions, which later turned out to hold for
regular primes. Kummer must have looked at FLT before, because in
a letter to Kronecker he said that "this time" he quickly found the
right approach.

The Paris Prize, as John Stilwell already wrote, did play a role for Kummer, as he confessed in one of his letters to Kronecker that can be found in Kummer's Collected Papers. But mathematically, Kummer attached importance only to the higher reciprocity laws.

Kummer worked out the arithmetic of cyclotomic extensions guided
by his desire to find the higher reciprocity laws; notions such
as unique factorization into ideal numbers, the ideal class group,
units, the Stickelberger relation, Hilbert 90, norm residues and
Kummer extensions owe their existence to his work on reciprocity laws.
His work on Fermat's Last Theorem is connected to the class number
formula and the "plus" class number, and a meticulous investigation
of units, in particular Kummer's Lemma, as well as the tools needed
for proving it, his differential logarithms, which much later were
generalized by Coates and Wiles. Some of the latter topics were helpful to Kummer later when he actually proved his higher reciprocity law.

Franz Lemmermeyer is better qualified than I am to answer
this question, and he certainly knows all about Kummer's motivation from
reciprocity laws. However, to answer some parts of your question: Kummer
certainly did publish papers on Fermat's last theorem, and he received a
prize from the Paris Academy in 1857 for his work on FLT.

He came under consideration for his long paper in the J. de Math. Pures
et Appl. of 1851. This was his first extended exposition of his work on
cyclotomic integers and ideal prime factorization, and he concluded it by
proving a special case of FLT. Thus, even if his real motivation was reciprocity
laws, he apparently thought that the best way to "sell" his theory (at
least to the French) was by an application to FLT.

I too defer to Franz's deep knowledge on such topics. Everything that I have read in both the primary and secondary literature completely agrees with what Franz has written here and elsewhere. Besides the link I gave to the discussion in his interesting paper on Jacobi and Kummer's ideal numbers, one may also find helpful the following passage from p. 15 of his beautiful book on reciprocity laws. It provides a concise summary of what we currently know about such matters. I quote it below since some readers may not have convenient access to the book.

The role of Fermat's Last Theorem in the development of algebraic
number theory is often overrated, probably due to Hensel's (false)
story claiming Kummer's first manuscript to have been an incorrect
proof of this problem. The crème de la crème of French mathematicians
- Lame, Legendre, Liouville, and Cauchy - tried their luck but didn't
really advance algebraic number theory during their work on FLT.
Gauss did not value it very highly, but admits that it made him take
up his investigations in number theory again: in a letter to Olbers
from March 21, 1816, he writes

I admit, that Fermat's Theorem, as an isolated result, has little
interest for me, since I can easily make a lot of such claims that
can be neither proved nor disproved. Nevertheless it made me take up
again some old ideas about a large extension of the higher arithmetic.
[...] Yet I am convinced, if luck should do more than I may expect
and if I succeed in making some of the main steps in that theory,
then Fermat's Theorem will appear as one of the less interesting
corollaries.

Gauss's last remark clearly indicates that he was at least thinking
about the arithmetic in cyclotomic number fields ${\mathbb Q}(\zeta_p)$, even
when, in a letter to Bessel a few months later, he reveals that the
investigations in question had to do with the part that he eventually
would publish, namely the theory of biquadratic residues. Parts of
his research were published in 1828 [272] and 1832 [273], and the
last paper contains the statement that cubic reciprocity is best
described in ${\mathbb Z}[\rho]$, where $\rho^2 + \rho + 1 = 0$, and that, more generally,
the study of higher reciprocity laws should be done after adjoining
higher roots of unity.

Even Kummer, who is responsible for the greatest advance towards of
Fermat's Last Theorem before the recent developments, got the main
motivation for studying cyclotomic fields from his desire to find a
general reciprocity law (which he called his "main enemy" in [Ku, Feb
25, 1848]). In almost every letter to Kronecker written between 1842
and 1848, Kummer mentions results related to reciprocity; the Fermat
equation is mentioned for the first time in [Ku, Apr. 02, 1847]. In
[Ku, Sept. 17, 1849] he informs Kronecker about the prize of 3000
Francs that the French Academy had offered to pay for a solution of
Fermat's Last Theorem, and in [Ku, Jan. 14, 1850] he writes

Once I will have fathomed whether this is so or not, then I will drop
the avaricious plans and work again only for the science, especially
for the reciprocity laws for which I have already envisaged some ideas.

It is therefore safe to say that it was the quest for higher
reciprocity laws that made Kummer study abelian extensions of $\mathbb Q$,
Eisenstein those of ${\mathbb Q}(i)$ and Hilbert those of general number fields.
Ironically, it was Hilbert himself who started the rumour that it was
FLT that led Kummer to his ideal numbers: in his famous address at
the ICM in Paris 1900, he wrote

stimulated by Fermat's problem, Kummer arrived at the introduction of
his ideal numbers and discovered the theorem of unique factorization
of the integers of cyclotomic fields into ideal prime factors.

Already in 1910, Hensel talked about "incontestable evidence" (actually it
was something that Gundelfinger had heard from H.G. Grassmann) for the
existence of a manuscript in which Kummer had claimed to have solved
Fermat's Last Theorem, a rumour eventually dismissed by Edwards [Ed1, Ed2]
and Neumann [Neu].

@Robin: thanks for posting the link. Alas, such links don't always work due to various factors (viewing limits, country restrictions, etc). In fact that page is currently blocked for me, and I doubt I've reached viewing limits since I have my own copy of the book. That's why I posted the excerpt.
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Bill DubuqueAug 7 '10 at 19:22